Operations Research Exercise 10.1

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 10.1 Business Maths Chapter 10 Operations Research for your reference and study.

You can find links to the other exercises below.

Solutions to Operations Research Exercise 10.2

Solutions to Operations Research Exercise 10.3

Solutions to Operations Research Exercise 10.4

Operations Research Exercise 10.1

$\text{1. What is transportation problem?}\\$
$\text{The objective of transportation problem is to determine the amount to be}\\$
$\text{transported from each origin to each destinations such that the total }\\$
$\text{transportation cost is minimized.}\\$

$\text{2. Write mathematical form of transportation problem.}\\$
$\text{Let there be m origins and n destinations. Let the amount of supply }\\$
$\text{at the }i^{th}\; \text{origin is}\; a_{i}.\; \text{Let the demand at}\; j^{th} \; \text{destination is} \; b_{j.}\\$
$\text{The cost of transporting one unit of an item from origin i to destination}\\$
$\text{ j is}\; c_{ij}\; \text{ and is known for all combinations (i,j). Quantity transported} \\$
$\text{from origin i to destination j is}\; x_{ij .\\$

$\text{The objective is to determine the quantity}\; x_{ij} \text{to be transported }\\$
$\text{ over all routes (i,j) so as to minimize the total transportation} \\$
$\text{ cost. The supply limits at the origins and the demand requirements } \\$
$\text{at the destinations must be satisfied.} \\$
$\text{The linear programming model representing the transportation problem is:}\\$
$\text{The objective function is Minimize Z =}\sum_{i=1}^{m} \sum_{j=1}^{n} C_{ij} x_{ij} \\$
$\text{Subject to the constraints}\\$
$\sum_{j=1}^{n}x_{ij} =a_{i}i=1,2....m \;\text{(Supply constraints)} \\$
$\sum_{i=1}^{m}x_{ij} =b_{j}i=1,2....n \;\text{(Demand constraints)} \\$
$x_{ij} \ge 0 \;\text{for all i,j }\text{(non-negative restrictions)} \\$

$\text{3. What is feasible solution and non degenerate solution in transportation problem?}\\$
$\text{A feasible solution to a transportation problem is a set of non-negative } \\$
$\text{values }x_{ij}\; \text{(i=1,2,..,m, j=1,2,…n) that satisfies the constraints.} \\$
$\text{If a basic feasible solution to a transportation problem contains exactly} \\$
$\text{ m+n-1 allocations in independent positions, it is called a Non degenerate } \\$
$\text{basic feasible solution. Here m is the number of rows and n is the number} \\$
$\text{ of columns in a transportation problem.} \\$

$\text{4. What do you mean by balanced transportation problem?}\\$
$\text{A balanced transportation problem is a transportation problem in which }\\$
$\text{the total supply exactly equals the total demand.}\\$

$\text{5. Find an initial basic feasible solution of the following}\\$
$\text{problem using north west corner rule.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$O_{1}\to D_{1}, \; O_{1}\to D_{2}, \; O_{2}\to D_{2}, \; \\$
$O_{2}\to D_{3}, \; O_{3}\to D_{3}, \; O_{3}\to D_{4}\\$
$\text{Total Transportation Cost}:=16\times 5+3\times \\$
$3+15\times 7+22\times 9+9\times 7+25\times 5\\$
$\text{Total Transportation Cost}:80+9+105+198+63+125\\$
$\text{Total Transportation Cost}:Rs. 580\\$

$\text{6. Determine an initial basic feasible solution of the following }\\$
$\text{transportation problem by north west corner method.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$\text{Chennai} \to \text{Bangalore}, \; \text{Madurai} \to \text{Bangalore},\\$
$\text{Madurai} \to \text{Nasik}, \; \text{Madurai} \to \text{Bhopal}, \; \text{Trichy} \\$
$\to \text{Bhopal}, \; \text{Trichy} \to \text{Delhi} \\$

$\text{Total Transportation Cost}:=30\times 6+5\times 5+28\times 11+7\times 9+25\times 7+25\times 13\\$
$\text{Total Transportation Cost}:180+25+308+63+175+325\\$
$\text{Total Transportation Cost}:Rs. 1076\\$

$\text{7. Obtain an initial basic feasible solution to the following}\\$
$\text{transportation problem by using least- cost method.} \\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$O_{1}\to D_{1}, \; O_{1}\to D_{3}, \; O_{2}\to D_{3}, \; O_{3}\to D_{1}, \; O_{3}\to D_{2}\\$
$\text{Total Transportation Cost}:=15\times 9+10\times 5+35\times 4+15\times 7+25\times 6\\$
$\text{Total Transportation Cost}:135+50+140+105+150\\$
$\text{Total Transportation Cost}:Rs. 580\\$

$\text{8. Explain Vogel's approximation method by obtaining initial feasible}\\$
$\text{solution of the following transportation problem.} \\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Thus, we have the following allocations.}\\$

$\text{Transportation Schedule}:\\$
$O_{1}\to D_{1}, \; O_{1}\to D_{2}, \; O_{2}\to D_{4}, \; O_{3}\to D_{1}, \; O_{3}\to D_{3}\; O_{3}\to D_{4}\\$
$\text{Total Transportation Cost}:=1\times 2+5\times 3+1\times 1+3\times 15+1\times 9\\$
$\text{Total Transportation Cost}:2+15+1+30+45+9\\$
$\text{Total Transportation Cost}:Rs. 102\\$

$\text{9. Consider the following transportation problem.}\\$

$\text{Determine initial basic feasible solution by VAM.} \\$

$\text{Total availability =Total requirement. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Thus, we have the following allocations.}\\$

$\text{Transportation Schedule}:\\$
$O_{1}\to D_{1}, \; O_{1}\to D_{3}, \; O_{2}\to D_{1}, \; O_{2}\to D_{2}, \; O_{2}\to D_{4}\; O_{3}\to D_{2}\\$
$\text{Total Transportation Cost}:=10\times 5+20\times 3+20\times 4+20\times 5+10\times 4+20\times 2\\$
$\text{Total Transportation Cost}:50+6080+100+40+40\\$
$\text{Total Transportation Cost}:Rs. 370\\$

$\text{10. Determine basic feasible solution to the following transportation}\\$
$\text{problem using North west Corner rule.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$P\to A, \; P\to B, \; Q\to B, \; Q\to C, \; Q\to D,\; R\to D, R\to E,\\$
$\text{Total Transportation Cost}:=3\times 2+1\times 11+2\times 4+4\times 7+2\times 2 3\times 8+6\times 12\\$
$\text{Total Transportation Cost}:6+11+8+28+4+24+72\\$
$\text{Total Transportation Cost}:Rs. 153\\$

$\text{11. Find the initial basic feasible solution of the following }\\$
$\text{transportation problem: Using North West Corner rule}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$A\to I, \; B\to I, \; B\to II, \; C\to II, \; C\to III\\$
$\text{Total Transportation Cost}:=7\times 1+3\times 0+1\times 1+10\times 5\\$
$\text{Total Transportation Cost}:7+0+36+1+50\\$
$\text{Total Transportation Cost}:Rs. 94\\$

$\text{12. Find the initial basic feasible solution of the following }\\$
$\text{transportation problem: Using Least Cost method.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$A\to III, \; B\to I, \; B\to III, \; C\to II, \; C\to III\\$
$\text{Total Transportation Cost}:=7\times 6+10\times 0+2\times 2+10\times 1+1\times 5\\$
$\text{Total Transportation Cost}:42+0+4+10+5\\$
$\text{Total Transportation Cost}:Rs. 61\\$

$\text{13. Find the initial basic feasible solution of the following }\\$
$\text{transportation problem: Using Vogel's approximation method.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Thus, we have the following allocations.}\\$

$\text{Transportation Schedule}:\\$
$A\to I, \; B\to I, \; B\to III, \; C\to I, \; C\to II\\$
$\text{Total Transportation Cost}:=7\times 1+2\times 0+10\times 2+1\times 3+10\times 1\\$
$\text{Total Transportation Cost}:7+0+20+3+10\\$
$\text{Total Transportation Cost}:Rs. 40\\$

$\text{14. Obtain the initial basic feasible solution to the following }\\$
$\text{transportation problem by North West Corner method.}\\$

$\text{Total supply=Total demand. Therefore the given problem is a balanced }\\$
$\text{transportation problem. Therefore, there exists a feasible solution.}\\$

$\text{Transportation Schedule}:\\$
$A\to D, \; A\to E, \; B\to E, \; B\to F, \; C\to F, C\to C\\$
$\text{Total Transportation Cost}:=200\times 11+50\times 13+175\times 18+125\times 14+150\times 13+250\times 10\\$
$\text{Total Transportation Cost}:2200+650+3150+1750+1950+2500\\$
$\text{Total Transportation Cost}:Rs. 12,200\\$

Operations Research Exercise 10.1

Operations Research Exercise 10.1

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Operations Research Exercise 10.1

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